import numpy as np
from pypop7.optimizers.es.es import ES
[docs]class VDCMA(ES):
"""Linear Covariance Matrix Adaptation (VDCMA).
Parameters
----------
problem : dict
problem arguments with the following common settings (`keys`):
* 'fitness_function' - objective function to be **minimized** (`func`),
* 'ndim_problem' - number of dimensionality (`int`),
* 'upper_boundary' - upper boundary of search range (`array_like`),
* 'lower_boundary' - lower boundary of search range (`array_like`).
options : dict
optimizer options with the following common settings (`keys`):
* 'max_function_evaluations' - maximum of function evaluations (`int`, default: `np.Inf`),
* 'max_runtime' - maximal runtime to be allowed (`float`, default: `np.Inf`),
* 'seed_rng' - seed for random number generation needed to be *explicitly* set (`int`);
and with the following particular settings (`keys`):
* 'sigma' - initial global step-size, aka mutation strength (`float`),
* 'mean' - initial (starting) point, aka mean of Gaussian search distribution (`array_like`),
* if not given, it will draw a random sample from the uniform distribution whose search range is
bounded by `problem['lower_boundary']` and `problem['upper_boundary']`.
* 'n_individuals' - number of offspring, aka offspring population size (`int`, default:
`4 + int(3*np.log(problem['ndim_problem']))`),
* 'n_parents' - number of parents, aka parental population size (`int`, default:
`int(options['n_individuals']/2)`).
Examples
--------
Use the optimizer to minimize the well-known test function
`Rosenbrock <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.es.vdcma import VDCMA
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 2,
... 'lower_boundary': -5*numpy.ones((2,)),
... 'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000, # set optimizer options
... 'seed_rng': 2022,
... 'mean': 3*numpy.ones((2,)),
... 'sigma': 0.1} # the global step-size may need to be tuned for better performance
>>> vdcma = VDCMA(problem, options) # initialize the optimizer class
>>> results = vdcma.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"VDCMA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
VDCMA: 5000, 7.116226375179302e-18
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/3e838zd5>`_ for more details.
Attributes
----------
mean : `array_like`
initial (starting) point, aka mean of Gaussian search distribution.
n_individuals : `int`
number of offspring, aka offspring population size.
n_parents : `int`
number of parents, aka parental population size.
sigma : `float`
final global step-size, aka mutation strength.
References
----------
Akimoto, Y., Auger, A. and Hansen, N., 2014, July.
Comparison-based natural gradient optimization in high dimension.
In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 373-380). ACM.
https://dl.acm.org/doi/abs/10.1145/2576768.2598258
See the official Python version from Prof. Akimoto:
https://gist.github.com/youheiakimoto/08b95b52dfbf8832afc71dfff3aed6c8
"""
def __init__(self, problem, options):
ES.__init__(self, problem, options)
self.options = options
self.c_factor = options.get('c_factor', np.maximum((self.ndim_problem - 5.0)/6.0, 0.5))
self.c_c, self.c_1, self.c_mu, self.c_s = None, None, None, None
self.d_s = None
self._v_n, self._v_2, self._v_, self._v_p2 = None, None, None, None
def initialize(self, is_restart=False):
self.c_c = self.options.get('c_c', (4.0 + self._mu_eff/self.ndim_problem)/(
self.ndim_problem + 4.0 + 2.0*self._mu_eff/self.ndim_problem))
self.c_1 = self.options.get('c_1', self.c_factor*2.0/(
np.square(self.ndim_problem + 1.3) + self._mu_eff))
self.c_mu = self.options.get('c_mu', np.minimum(1.0 - self.c_1, self.c_factor*2.0*(
self._mu_eff - 2.0 + 1.0/self._mu_eff)/(np.square(self.ndim_problem + 2.0) + self._mu_eff)))
self.c_s = self.options.get('c_s', 1.0/(2.0*np.sqrt(self.ndim_problem/self._mu_eff) + 1.0))
self.d_s = self.options.get('d_s', 1.0 + self.c_s + 2.0*np.maximum(0.0, np.sqrt(
(self._mu_eff - 1.0)/(self.ndim_problem + 1.0)) - 1.0))
d = np.ones((self.ndim_problem,)) # diagonal vector of sampling distribution
# set principal search direction (vector) of sampling distribution
v = self.rng_optimization.standard_normal((self.ndim_problem,))/np.sqrt(self.ndim_problem)
p_s = np.zeros((self.ndim_problem,)) # evolution path for step-size adaptation (MCSA)
p_c = np.zeros((self.ndim_problem,)) # evolution path for covariance matrix adaptation (CMA)
self._v_n = np.linalg.norm(v)
self._v_2 = np.square(self._v_n)
self._v_ = v/self._v_n
self._v_p2 = np.square(self._v_)
z = np.empty((self.n_individuals, self.ndim_problem)) # Gaussian noise for mutation
zz = np.empty((self.n_individuals, self.ndim_problem)) # search directions
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
return d, v, p_s, p_c, z, zz, x, mean, y
def iterate(self, d=None, z=None, zz=None, x=None, mean=None, y=None, args=None):
for k in range(self.n_individuals):
if self._check_terminations():
return z, zz, x, y
z[k] = self.rng_optimization.standard_normal((self.ndim_problem,))
zz[k] = d*(z[k] + (np.sqrt(1.0 + self._v_2) - 1.0)*(np.dot(z[k], self._v_)*self._v_))
x[k] = mean + self.sigma*zz[k]
y[k] = self._evaluate_fitness(x[k], args)
return z, zz, x, y
def _p_q(self, zz, w=0):
zz_v_ = np.dot(zz, self._v_)
if isinstance(w, int) and w == 0:
p = np.square(zz) - self._v_2/(1.0 + self._v_2)*(zz_v_*(zz*self._v_)) - 1.0
q = zz_v_*zz - ((np.square(zz_v_) + 1.0 + self._v_2)/2.0)*self._v_
else:
p = np.dot(w, np.square(zz) - self._v_2/(1.0 + self._v_2)*(zz_v_*(zz*self._v_).T).T - 1.0)
q = np.dot(w, (zz_v_*zz.T).T - np.outer((np.square(zz_v_) + 1.0 + self._v_2)/2.0, self._v_))
return p, q
def _update_distribution(self, d=None, v=None, p_s=None, p_c=None, zz=None, x=None, y=None):
order = np.argsort(y)[:self.n_parents]
# update mean
mean = np.dot(self._w, x[order])
# update global step-size
z = np.dot(self._w, zz[order])/d
z += (1.0/np.sqrt(1.0 + self._v_2) - 1.0)*np.dot(z, self._v_)*self._v_
p_s = (1.0 - self.c_s)*p_s + np.sqrt(self.c_s*(2.0 - self.c_s)*self._mu_eff)*z
p_s_2 = np.dot(p_s, p_s)
self.sigma *= np.exp(self.c_s/self.d_s*(np.sqrt(p_s_2)/self._e_chi - 1.0))
# update restricted covariance matrix (d, v)
h_s = p_s_2 < (2.0 + 4.0/(self.ndim_problem + 1.0))*self.ndim_problem
p_c = (1.0 - self.c_c)*p_c + h_s*np.sqrt(self.c_c*(2.0 - self.c_c)*self._mu_eff)*np.dot(self._w, zz[order])
gamma = 1.0/np.sqrt(1.0 + self._v_2)
alpha = np.sqrt(np.square(self._v_2) + (1.0 + self._v_2)/np.max(self._v_p2)*(2.0 - gamma))/(2.0 + self._v_2)
if alpha < 1.0:
beta = (4.0 - (2.0 - gamma)/np.max(self._v_p2))/np.square(1.0 + 2.0/self._v_2)
else:
alpha, beta = 1.0, 0.0
b = 2.0*np.square(alpha) - beta
a = 2.0 - (b + 2.0*np.square(alpha))*self._v_p2
_v_p2_a = self._v_p2/a
if self.c_mu == 0:
p_mu, q_mu = np.zeros((self.ndim_problem,)), np.zeros((self.ndim_problem,))
else:
p_mu, q_mu = self._p_q(zz[order]/d, self._w)
if self.c_1 == 0:
p_1, q_1 = np.zeros((self.ndim_problem,)), np.zeros((self.ndim_problem,))
else:
p_1, q_1 = self._p_q(p_c/d)
p = self.c_mu*p_mu + h_s*self.c_1*p_1
q = self.c_mu*q_mu + h_s*self.c_1*q_1
if self.c_mu + self.c_1 > 0:
r = p - alpha/(1.0 + self._v_2)*((2.0 + self._v_2)*(
q*self._v_) - self._v_2*np.dot(self._v_, q)*self._v_p2)
s = r/a - b*np.dot(r, _v_p2_a)/(1.0 + b*np.dot(self._v_p2, _v_p2_a))*_v_p2_a
ng_v = q/self._v_n - alpha/self._v_n*((2.0 + self._v_2)*(
self._v_*s) - np.dot(s, np.square(self._v_))*self._v_)
ng_d = d*s
up_factor = np.minimum(np.minimum(1.0, 0.7*self._v_n/np.sqrt(np.dot(ng_v, ng_v))),
0.7*(d/np.min(np.abs(ng_d))))
else:
ng_v, ng_d, up_factor = np.zeros((self.ndim_problem,)), np.zeros((self.ndim_problem,)), 1.0
v += up_factor*ng_v
d += up_factor*ng_d
self._v_n = np.linalg.norm(v)
self._v_2 = np.square(self._v_n)
self._v_ = v/self._v_n
self._v_p2 = np.square(self._v_)
return mean, p_s, p_c, v, d
def restart_reinitialize(self, d=None, v=None, p_s=None, p_c=None, z=None, zz=None, x=None, mean=None, y=None):
if self.is_restart and ES.restart_reinitialize(self, y):
d, v, p_s, p_c, z, zz, x, mean, y = self.initialize(True)
return d, v, p_s, p_c, z, zz, x, mean, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
d, v, p_s, p_c, z, zz, x, mean, y = self.initialize()
while not self.termination_signal:
# sample and evaluate offspring population
z, zz, x, y = self.iterate(d, z, zz, x, mean, y, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
mean, p_s, p_c, v, d = self._update_distribution(d, v, p_s, p_c, zz, x, y)
self._n_generations += 1
d, v, p_s, p_c, z, zz, x, mean, y = self.restart_reinitialize(
d, v, p_s, p_c, z, zz, x, mean, y)
results = self._collect(fitness, y, mean)
results['d'] = d
results['v'] = v
return results